Asymptotic and numerical modelling of flows in fractured porous media

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fractured porous media

Philippe Angot, Franck Boyer, Florence Hubert

To cite this version:

Philippe Angot, Franck Boyer, Florence Hubert. Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2009, 23 (2), pp.239–275. �hal-00127023v2�

IN FRACTURED POROUS MEDIA

Philippe ANGOT, Franck BOYER and Florence HUBERT

Abstract. This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale, so that we can asymptotically reduce them to immersed polygonal fault interfaces and the model finally consists in a coupling between a 2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures.

A cell-centered finite volume scheme on general polygonal meshes fitting the interfaces is derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocity jumps through the interfaces. We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures. Some numerical results are reported showing different kinds of flows in the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numerical solutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.

1991 Mathematics Subject Classification. 76S05 - 74S10 - 35J25 - 35J20 - 65N15.

The dates will be set by the publisher.

1. Introduction

The present work addresses the numerical modelling of monophasic flows in saturated fractured porous media by means of finite volume methods. The flow in the fracture domain Ωf, in general fully immersed in the porous medium Ω, is assumed to be governed by the Darcy law, as is the flow in the porous matrix, with an anisotropic permeability tensor Kf. Our objective is to study asymptotic “double-permeability” models of fracture flow interacting with the matrix flow where the fractures are reduced to sharp interfaces Σ when the fracture aperture bf goes to zero. More precisely, if lm and lp denote respectively the macroscopic and pore length scales, we have : lp ≪bf ≪lm. The models involve some algebraic or differential immersed transmission conditions on the mean fracture surface Σ which combine the jumps of both pressure and normal velocity through the fault interface. The fractures may be “impermeable” (no jump of normal velocity with jumps of pressure on Σ),

“highly permeable” (jumps of normal velocity with no jump of pressure on Σ), or characteristic of intermediate

Keywords and phrases: Fractured porous media, Darcy flow, Finite volume method, Asymptotic models of flow.

1 Universit´e de Provence and CNRS, Laboratoire d’Analyse Topologie et Probabilit´es 39 rue F. Joliot Curie, F-13453 Marseille cedex 13.

Email : [angot,fboyer,fhubert]@cmi.univ-mrs.fr

c

∞l∞∋EDP Sciences, SMAI 1999

cases according to some physical phenomena altering their properties with time : mechanical effects (erosion, sedimentation, clogging, thermomechanical stress), chemical effects... [6, 18].

The global Darcy simulations of such problems, typically requiring refined meshes inside the fracture domain Ωf, can be very expensive. Thus, asymptotic models where the fractures are reduced to immersed polygonal fault interfaces, are useful to provide a good approximation of the global flow at a lower cost.

There is a very large literature on the problem of modelling flows in fractured porous media both in the monophasic and multiphasic cases. Reviews on such problems can be found in [1, 6, 7, 18], for instance. Some authors neglect the flow in the porous matrix and only concentrate on the study of the flow in fracture networks (see [1, 7] and references therein), others propose to treat the specific geometry of the fractures by a specific numerical method (see for instance [12, 13] where joint elements are used). In the diphasic case, where a convection-diffusion-dispersion equation has also to be taken into account, we can refer to the recent reference [24] in which the authors solve their model by using a cell-centered finite volume method.

In the single-phase case we are interesed in, our approach follows for instance [17, 21, 22] and consists in writing an asymptotic model directly on the continuous problem and then to propose an adapted numerical method to treat the obtained simplified system of equations. More precisely, a similar model than the one we propose in section 2 was studied in [17, 22] only in the particular case in which the fracture interface Σ is not immersed inside the domain Ω, but separates it into disjoint subdomains. We propose in this paper thee extension of this model to the case of fully immersed fractures and we propose and analyse a corresponding numerical scheme. Our analysis will also be valid for a more general range of values for a quadrature parameter ξwhich appears in the model (see Section 2.2.5 for a detailed discussion on this point).

Outline

The paper is organized as follows. Section 2 is devoted to the presentation of the asymptotic models for flow in 2D fractured porous media we are interested in. The permeability anisotropy along the curvilinear coordinates associated with a fracture Σ is taken into account. The models depend on some quadrature rules used to approximate the mean of the variables calculated transversely to the fracture and are characterized by a real parameterξ≥¹₂.

We state and prove in Section 3 the global solvability of the asymptotic models in the case of a fully immersed fracture inside the porous matrix and for any value of the parameter ξ in the range [1/2,+∞[. In Section 4, a cell-centered finite volume scheme is proposed to approximate the solution of this problem. We prove the convergence of the finite volume approximate solution towards the unique solution of the asymptotic model under study, for any value of the parameterξ≥1/2.

Numerical investigations of the validity of the asymptotic models are proposed in Section 5. All the numerical results obtained through the asymptotic models are compared to the solutions of the global double-permeability Darcy system. In order to obtain these reference solutions we use a modified Discrete Duality Finite Volume scheme on meshes which are locally refined in the neighborhood of the fractures. Such m-DDFV schemes were proved to be first-order in the discrete H¹- norm in [11], for any kind of anisotropy and heterogeneity of the permeability tensor.

We first illustrate the various typical flows we can expect depending on the physical properties of the fracture (impermeable, permeable, ...) in the case of a single fracture and we study the influence of the choice of the quadrature parameterξ.

Then, the behavior of the model and some of its limits are illustrated for more complex situations: a fracture network and a non-constant aperture fracture. Finally, we compare our results with analytical solutions obtained in [23] in the particular case of a lens-shaped fracture in an infinite porous matrix.

2. Global and asymptotic models for flows in 2D fractured porous media

In the whole paper we consider an open bi-dimensional polygonal bounded domain ˜Ω⊂R² representing the porous medium under study. We are interested in the saturated flow of a single incompressible fluid in such a

porous medium described by the mass conservation equation and the Darcy law. Hence, the main unknowns of the problem are the pressure field pand the related filtration velocityv^p.

The boundary Γ^def= ∂Ω is divided into two disjoint subsets Γ = Γ˜ D∪ΓN, ΓD∩ΓN =∅, corresponding to Dirichlet boundary conditions (the pressure is prescribed on ΓD) and to Neumann boundary condition (mass flux is imposed to be 0 on ΓN). It is of course possible to consider more general boundary conditions on ΓN like non-homogeneous Fourier boundary conditions for instance but we focus our attention here on homogeneous Neumann conditions.

2.1. The global Darcy model and its geometry

We first recall the standard global Darcy model in the case where ˜Ω is a fractured porous medium. Without any loss of generality we will assume in the analysis that:

• there is a unique fracture in the porous medium. Multiple disjoint fractures can be treated in the very same way. The case of multiple crossing fractures would require some natural supplementary conditions (continuity of the pressure and mass flux balance equation) at the crossing points that we do not tackle in this work.

• the fracture is fully immersed in ˜Ω. As we will see in the numerical results (see section 5), the case where some part of the fracture is in contact with the boundary of the domain can also be considered without additional difficulties.

The geometry is represented in the left part of Figure 1. The porous medium ˜Ω is split into two subdomains:

the porous matrix Ωmand the fracture Ωf. We assume that the fracture domain Ωf has the following form Ωf =

s+tν(s)/ s∈Σ, t∈

−bf(s) 2 ,bf(s)

2

, (1)

where Σ is a 1D polygonal broken line without self-intersection. For any s ∈ Σ, ν(s) denotes a unit normal vector to Σ at the points(its orientation has no importance at that point and will be precised later), andbf(s) denotes the fracture aperture at the points. Throughout the paper we will assume that

bf ∈ C¹(Σ), inf

Σ bf >0. (2)

The main physical assumptions we consider are:

• The permeability tensorKmin the porous matrix Ωmis constant and isotropic. Since it is isotropic we will often considerKmas a positive real number.

• The permeability tensorKf is constant and anisotropic in the fracture. More precisely, we assume that, in the curvilinear frame (τ,ν) linked to Σ, we have

Kf =

Kf,τ 0 0 Kf,ν

, (3)

that is (Kfτ,τ) = Kf,τ, (Kfν,ν) =Kf,ν. Without any additional work it is possible to take into account smooth variations ofKf,τ andKf,νwith respect to the tangential variablesin Ωf, but we will assume here for simplicity that Kf is constant.

Note that we assume thatKf is diagonal in the curvilinear frame, that is (Kfτ,ν) = 0. This corre- sponds to many physical situations since the fracture orientation is very likely to be also a characteristic direction of the pore structures inside Ωf. Nevertheless, it is possible to write down a model for general permeabilities such that (Kfτ,ν) 6= 0 but its analysis is then much more intricate. Notice that, in practice, the coefficients Kf,τ and Kf,ν can be estimated by specific studies of flows inside different kinds of fractures (see [1]).

With these assumptions at hand, the pressure distribution in the medium is solution to the following global Darcy problem

∇ ·v^p=Qin ˜Ω, (PG-a)

v^p=−1

µK(x)· ∇pin ˜Ω, (PG-b)

p=p^D on ΓD, (PG-c)

v^p·n= 0 on ΓN, (PG-d)

whereK(x) =Kmforx∈ΩmandK(x) =Kf forx∈Ωf,µ >0 is the viscosity of the fluid,Qa mass source term and p^D the Dirichlet boundary data for the pressure. Notice that, without any loss of generality, we do not take into account the gravity terms in this problem which leads us to simplified notation throughout the paper.

From now on, we assume that the fracture aperturebf is small compared to the length scale of the fracture, that isbf ≪ |Σ|. Furthermore, we emphasize the fact that the permeability coefficients inside Ωf can be very different from the porous matrix permeabilityKm. These properties imply that problem (PG) may be difficult to solve numerically. Our objective is to propose in the sequel an asymptotic model in a simplified geometry aiming at providing a good approximation of the flow in the fractured medium in the considered situation.

Darcy law in the porous matrix Darcy law in the porous matrix

Darcy law in the fracture 1D Darcy law in the fracture

+ jump conditions Ω⁻

∂Σ⁺ τ

ν=n⁻=−n⁺

∂Σ⁻

permeabilityK_m n

Ω⁺ permeabilityK_m

permeabilityK_f b_f

Σ Ωf

Ωm

Γ

Σ˜

Geometry of the asymptotic Darcy model Geometry of the global Darcy model

Figure 1. Configuration for “‘double-permeability” models.

2.2. The asymptotic model and its geometry

We recall here a possible formal derivation of the asymptotic model for flows in such a fractured porous medium we are interested in.

The main idea is to reduce the 2D fracture domain Ωf inside ˜Ω by the 1D polygonal broken line Σ. As a consequence, the porous matrix domain Ωmwill be formally replaced by the (larger) open set Ω = ˜Ω\Σ. Notice that ˜Ω = Ω∪Σ and that∂Ω = Γ∪Σ. Note that Ω is a domain with cuts. In particular, does not lie locally on one side of its boundary near the fracture, which will introduce some technical difficulties (see section 3.1.1).

Let us now give the main notation we will use in the sequel.

2.2.1. Notation

• We first embed Σ within a larger polygonal broken line ˜Σ⊃Σ which divides the domain ˜Ω into two open disjoint subdomains Ω⁻ and Ω⁺such that ˜Ω = Ω⁻∪Σ˜∪Ω⁺, see the right part of Figure 1. Notice that the sets ˜Σ, Ω⁺ and Ω⁻ will never appear either in the final set of equations or in the numerical scheme we will propose. They are only needed to fix an orientation and to let us give a precise meaning of some trace operators on Σ in section 3.1.1.

• Letnbe the outward unit normal vector on the boundary Γ of the computational domain, andν the unit normal vector on Σ oriented from Ω⁻ to Ω⁺. The outward unit normal to ∂Ω⁺ on Σ is then n⁺=−ν and the outward unit normal to ∂Ω⁻ on Σ isn⁻=ν. Letτ be a unit tangential vector on Σ so that (τ,ν) is positively oriented, and s a normalized curvilinear coordinate parametrizing Σ in the direction given byτ. In our bidimensional situation, the boundary of Σ is composed of two points

∂Σ ={∂Σ⁺, ∂Σ⁻}defined in such a way thats= 0 in∂Σ⁻ ands=|Σ|in∂Σ⁺.

• For any functionψin H¹(Ω), let γ⁺ψ andγ⁻ψ be the traces ofψon each side of Σ (see Section 3.1.1 for precise definitions),ψ= ¹₂(γ⁺ψ+γ⁻ψ) be their arithmetic mean, and [[ψ]] = (γ⁺ψ−γ⁻ψ) be their jump across Σ oriented byν. Let ∇τ and∇τ·denote the tangential gradient and divergence operators along Σ.

2.2.2. Governing equations for the flow in the porous matrix

The first part of this model consists in writing the isotropic homogeneous Darcy law inside the new larger porous matrix domain Ω :

∇ ·v^p = Qin Ω (4)

v^p = −1

µKm· ∇pin Ω (5)

p = p^D on ΓD (6)

v^p·n = 0 on ΓN. (7)

We want to emphasize the fact that, contrary to the global Darcy model (PG), this system is posed on the domain Ω, and that the permeability tensorKmis constant and isotropic. In this set of equations, the properties of the fracture Σ are not yet taken into account and of course the system is not closed since we need to prescribe the behavior of the solution on Σ. This will be done in the following section.

2.2.3. Averaging the Darcy law across the fracture

We describe in this section the way to formally derive a supplementary set of equations posed on the fracture Σ which will complete the problem (4)-(7) leading to a well-posed problem which is supposed to be a satisfactory approximation of the solution to the initial global Darcy model.

We recall that the flow inside the fracture domain Ωf (whose geometry is given in (1)) is described by the Darcy law for a permeability tensor Kf given in the curvilinear frame by (3).

Transversely to Σ, we define the following mean quantities of the variables :

uf,τ(s) = 1 bf(s)

Z ^bf₂^(s)

−^bf₂^(s)

v^p(s, t)·τ(s)dt, uf,ν(s) = 1 bf(s)

Z ^bf₂^(s)

−^bf₂^(s)

v^p(s, t)·ν(s)dt,

Πfp(s) = 1 bf(s)

Z ^bf₂^(s)

−^bf₂^(s)

p(s, t)dt, Qf(s) = 1 bf(s)

Z ^bf₂^(s)

−^bf₂^(s)

Q(s, t)dt, wherepandv^p are solutions to the global Darcy model (PG).

First, we average the mass conservation equation (PG-a) and the Darcy law (PG-b) over the cross-section of the fracture to obtain :

∇τ ·(bfuf,τ) + [[v^p·ν]] = bfQf, uf,τ = −1

µKf,τ∇τΠfp, uf,ν = −1

µKf,ν

[[p]]

bf

.

Then, by using the trapezoidal quadrature rule to approximate the mean variables with an error ofO(b²_f/|Σ|²), that is

Πfp≃p, uf,ν≃v^p·ν, (8)

we get the first asymptotic model of flow along the fault interface Σ :

∇τ ·(bfuf,τ) = bfQf−[[v^p·ν]] in Σ (9) uf,τ = −1

µKf,τ∇τΠfp, in Σ (10)

Πfp = p, in Σ, (11)

v^p·ν = −1 µKf,ν

[[p]]

bf

, in Σ, (12)

where, in these equations the jumps and the mean-values of pand v^p·ν are now the one obtained from the solution of the asymptotic model of flow inside the approximate porous matrix Ω, that is system (4)-(7).

Finally, we need to close the system with a condition on the boundary ∂Σ of the fracture. We will consider a Neumann boundary condition

uf,τ = 0 on∂Σ.

This boundary condition states that, since the fracture aperture is small, the mass transfer across the extremities of the fracture can be neglected in front of the transversal one. Other kinds of boundary conditions can of course be considered: in the case where the fracture is touching the exterior boundary ΓD, it can be natural to impose a Dirichlet boundary condition on the pressure on ∂Σ. We will see that these boundary conditions are not always acurate and should be replaced by more physical conditions (see some numerical results in Section 5.4).

2.2.4. Generalization for other quadrature rules

Such models have already been proposed in [17, 21]. In fact, in those references other quadrature rules are used in place of the trapezoidal rule to approximate the cross-section mean values of the pressure Πfpand of v^p·ν in (8). Following the computations in the above references, we may replace (11) by

Πfp=p+(2ξ−1)µ 4Kf,ν

bf[[v^p·ν]], on Σ, whereξ≥1/2 is a quadrature parameter.

For example, the trapezoidal rule (11) is recovered whenξ= 1/2 which appears to be the most natural and simplest choice, whereas the use of the mid-point rule givesξ = 3/4. We give a numerical comparison of the models for various values of this parameter in Section 5.

2.2.5. The final asymptotic model under study

Gathering the previous equations, we finally obtain the following system of equations that we will analyse in the sequel of the paper:

∇ ·v^p=Qin Ω (PA-a)

v^p=−1

µKm· ∇pin Ω (PA-b)

p=p^D on ΓD (PA-c)

v^p·n= 0 on ΓN, (PA-d)

∇τ ·(bfuf,τ) =bfQf−[[v^p·ν]] in Σ, (PA-e) uf,τ =−Kf,τ

µ ∇^τΠfpin Σ, (PA-f)

uf,τ = 0, on∂Σ, (PA-g)

v^p·ν=−Kf,ν

µ [[p]]

bf in Σ, (PA-h)

Πfp=p+(2ξ−1)µ 4Kf,ν

bf[[v^p·ν]] in Σ. (PA-i) The quadrature parameterξ∈₁

2,+∞

appearing in (PA-i) is now fixed throughout the paper.

In [21, 22] a similar model is studied in the case where Σ is not immersed inside the domainΩ. Furthermore,e their theoretical and numerical analysis are valid for values of the quadrature parameter ξ > 1/2 (typically ξ= 3/4 or ξ= 1) so that, in this reference, the most natural model (that is when ξ= 1/2) is not taken into account. This is due to the fact that the casesξ= 1/2 andξ >1/2 have a very different mathematical structure (we will see in the sequel that the proofs are often different in the two cases). Hence, the mixed formulation used in these references did not allow to perform the analysis forξ= 1/2. Note that the model forξ= 3/4 is also numerically studied in [17] in the case of an isotropic fracture permeability tensorKf. For the case of a fully immersed fracture, the asymptotic model is numerically investigated in [10] forξ= 3/4.

In [3–5], it is also proposed (in the caseξ= 1/2) to replace the partial differential equation (PA-e)-(PA-g) on the fracture by a simpler algebraic model. More precisely, in these references, equations (PA-a)-(PA-d) and (PA-h)-(PA-i) are conserved whereas (PA-e)-(PA-g) are replaced by

[[v^p·ν]] =−bfKf,τ

µ 1

s(p−P)

+bfQf on Σ,

whereP is a given reference pressure ats= 0. This leads to a simpler but less precise model.

3. Well-posedness of the asymptotic models

We denote byk · k0,Ωthe L²−norm on Ω and by k · k1,Ω theH¹−norm on Ω. For anyU ⊂∂Ω, let L²(U), H¹²(U) andH¹(U) be the standard Lebesgue and Sobolev spaces onU endowed with their respective standard normsk · k0,U, k · k_1/2,U andk · k1,U.

3.1.1. Trace results for fractured domains

We define H_Γ¹(Ω) = {p ∈ H¹(Ω), p = 0 on Γ}. Let us state the trace results available for the fractured domain Ω. We denote byγ0,Dandγ0,N the trace operators fromH¹(Ω) ontoH¹²(ΓD) andH¹²(ΓN) respectively.

Since the fracture Σ is supposed to be immersed in Ω, these operators are classicaly defined in the same way than in the case of a smooth domain.

We concentrate now on the trace problem on the fracture Σ. First of all, we define the two linear and continuous trace operators γ⁺ and γ⁻ from H¹(Ω) on H¹²(Σ), by restricting to Σ the values of the standard trace operators γ⁺ and γ⁻ defined for the two domains Ω⁺ and Ω⁻, respectively. This definition does not depend on the way Ω⁺ and Ω⁻ are constructed from the interface Σ.

The spaceH₀₀¹²(Σ) is the set of the functionsg∈H¹²(Σ) such that _s(|Σ|−s)^|g(s)|2 ∈L¹(Σ) and is endowed with the norm defined bykgk²_1/2,00,Σ=kgk²_1/2,Σ+R|Σ|

0

|g(s)|²

s(|Σ|−s)ds.We denote byH⁻¹²(Σ) the dual space ofH¹²(Σ) and by ˜H⁻¹²(Σ) the dual space of H₀₀¹²(Σ).

Following [19, Th. 1.5.2.3, Th. 1.5.1.3], the techniques in [19, Section 1.7] and also [8, 9], applied to the open set Ω (whose boundary is polygonal with some angles of 2π), we have the following result.

Proposition 3.1. The global trace operatorγΣ onΣdefined by

γΣ:p∈H¹(Ω)7→γΣ(p) = (γ⁺(p), γ⁻(p))∈H¹²(Σ)×H¹²(Σ), is continuous from H¹(Ω) onto the space

TΣ= (

(g⁺, g⁻)∈

H¹²(Σ)2

, Z |Σ|

0

|g⁺(s)−g⁻(s)|²

s(|Σ| −s) ds <+∞

)

=

(g⁺, g⁻)∈

H¹²(Σ)2

, g⁺−g⁻∈H₀₀¹²(Σ)

,

endowed with the norm

k(g⁺, g⁻)kTΣ = kg⁺k²_1/2,Σ+kg⁻k²_1/2,Σ+ Z |Σ|

0

|g⁺(s)−g⁻(s)|² s(|Σ| −s) ds

!¹₂ .

Furthermore, there exists a continuous linear operator RΣ:TΣ→H_Γ¹(Ω) which is a right inverse of the global trace operator, that is

γΣ◦RΣ= IdTΣ. Finally, C_c^∞(Ω)is dense in kerγΣ∩H_Γ¹(Ω).

The proofs of the various statements of this proposition are contained in the references given above, except for the final density statement. For sake of completeness, we provide a proof of this statement in Appendix A.

With this result at hand, one can define the normal traces on Σ of any vector field inv∈Hdiv(Ω)^def={u∈ (L²(Ω))^d, ∇ ·u∈L²(Ω)}as an element of the dual space TΣ′ as follows.

Proposition 3.2. For any v∈Hdiv(Ω), the map v·n∈TΣ′ defined by g= (g⁺, g⁻)∈TΣ7→(v·n)(g)^def=

Z

Ω

v· ∇RΣ(g)dx+ Z

Ω

(∇ ·v)RΣ(g)dx,

is linear continuous and does not depend on the choice of the right inverse operator RΣ. Furthermore, there exists a unique element [[v·ν]]in H⁻¹²(Σ) and a unique elementv·ν in H˜⁻¹²(Σ) such that

hv·n, giTΣ′,TΣ =−

[[v·ν]],g⁺+g⁻ 2

H⁻¹²(Σ),H¹²(Σ)

− hv·ν,(g⁺−g⁻)i

H˜⁻¹²(Σ),H

1 002(Σ).

Proof. The continuity of the map v·nimmediately follows from the continuity of the operator RΣ and the definition of the norm inHdiv(Ω).

By definition of∇ ·v, for any φ∈ C_c^∞(Ω) we have Z

Ω

v· ∇φ dx+ Z

Ω

(∇ ·v)φ dx= 0.

This formula is still true for anyφ∈kerγΣ∩H_Γ¹(Ω) by using the density property in Proposition 3.1. Hence, the definition ofv·ndoes not depend on the right inverse operatorRΣwe choose.

We can now define [[v·ν]], andv·ν as follows h[[v·ν]], ψi

H⁻¹²(Σ),H¹²(Σ)=hv·n,(−ψ,−ψ)i_TΣ′,TΣ, ∀ψ∈H¹²(Σ), hv·ν, ψi

H˜⁻¹²(Σ),H

12 00(Σ)=

v·n,

−ψ 2,ψ

2

TΣ′,TΣ

, ∀ψ∈H₀₀¹²(Σ).

It is straightforward to see that the required properties hold.

Whenvis smooth enough (sayv∈(H¹(Ω))^d), [[v·ν]] and v·ν are respectively equal to the jump and the mean-value ofv·νacross Σ.

Finally, for anyφ∈H¹(Ω) and anyv∈Hdiv(Ω), we have the following Stokes-type formula Z

Ω

v·∇φ dx+

Z

Ω

(∇·v)φ dx=hv·n, γΓφi

H⁻¹2(Γ),H¹2(Γ)−h[[v·ν]], φi

H⁻¹2(Σ),H¹2(Σ)−hv·ν,[[φ]]i

H˜⁻¹²(Σ),H

1

002(Σ). (13) We conclude this section by a density result which will be useful in the sequel, in order to show the convergence of the numerical scheme. The idea is that the set of smooth functions in Ω, constant near the extremities of Σ but not necessarily continuous across Σ is dense in H¹(Ω).

Proposition 3.3. The space S defined by

S ={u∈ C^∞(Ω), s.t.u_|Ω¯^± ∈ C^∞( ¯Ω^±),anduis constant near the extremities ofΣ}, (14) is dense in H¹(Ω).

This result can be proved following similar lines than the proof of the density statement in Proposition 3.1 which is given in Appendix A.

3.1.2. Functional spaces

For any pressure field q ∈ H¹(Ω) defined inside the porous matrix, let us associate the Darcy velocity v^q ∈(L²(Ω))² defined by

v^q =−Km

µ ∇q, (15)

and, in the case wherev^q ∈Hdiv(Ω), we define the fracture pressure Πfqon Σ by Πfq=q+(2ξ−1)µ

4Kf,ν

bf[[v^q·ν]]∈H⁻¹²(Σ). (16) Notice that the productbf[[v^q·ν]] is well defined sincebf is supposed to be smooth (see (2)). We introduce the space

W=

q∈H¹(Ω) such that (2ξ−1)v^q ∈Hdiv(Ω),(2ξ−1)[[v^q·ν]]∈L²(Σ),Πfq∈H¹(Σ) ,

endowed with the norm

kqkW= kqk²_1,Ω+kΠfqk²_1,Σ+ (2ξ−1)k[[v^q·ν]]k²_0,Σ¹₂ .

Remark 3.4. Whenξ= 1/2, this space reduces to W ={q∈H¹(Ω),Πfq ∈H¹(Σ)} with the corresponding norm and, in that case,W is an Hilbert space, which is not the case whenξ >1/2.

3.2. Well-posedness of the problem (PA)

From now on,p^D∈H¹²(ΓD) is a given boundary data for the pressure. We calla solution of the asymptotic model, any function p∈ W that satisfies

∇ ·v^p = Qin Ω, (17)

p = p^D on ΓD, (18)

v^p·n = 0 on ΓN, (19)

−∇τ·

bfKf,τ

µ ∇τΠfp

= bfQf−[[v^p·ν]] in Σ, (20)

−Kf,τ

µ ∇τΠfp = 0 on∂Σ, (21)

v^p·ν = −Kf,ν

µ [[p]]

bf

on Σ, (22)

where v^p and Πfpare defined in (15)-(16) above. Equations (20) and (21) are required to be satisfied in the weak sense, that is inH⁻¹(Σ) andH^−1/2(∂Σ) respectively.

Our first result is the following.

Theorem 3.5. For any ξ≥ ¹₂, the problem (17)-(22)admits a unique solutionp∈ W.

Proof.

• Existence: This will be proved in the following section by passing to the limit in the finite volume scheme.

• Uniqueness: The problem being linear, it is enough to show that if p ∈ W is a solution to the homogeneous problem

v^p = −Km

µ ∇p, in Ω, (23)

∇ ·v^p = 0 in Ω, (24)

p = 0, on ΓD, (25)

v^p·n = 0 on ΓN, (26)

−∇τ·

bf

Kf,τ

µ ∇τΠfp

= −[[v^p·ν]] on Σ, (27)

−Kf,τ

µ ∇τΠfp = 0 on∂Σ, (28)

v^p·ν = −Kf,ν

µ [[p]]

bf

on Σ, (29)

then we havep= 0.

To this end, we usepas a test function in (24), and using the Stokes formula (13), we get

0 =

Z

Ω

p∇ ·v^pdx

= Z

Ω

Km

µ |∇p|²dx− h[[v^p·ν]], pi

H⁻¹²(Σ),H¹²(Σ)− hv^p·ν,[[p]]i

H˜⁻¹²(Σ),H

1 2 00(Σ)

= Z

Ω

Km

µ |∇p|²dx− h[[v^p·ν]], pi

H⁻¹2(Σ),H¹2(Σ)+ Z

Σ

Kf,ν

µ

[[p]]

bf(s)

2

bf(s)ds.

Sincep∈ W, we can use Πfp∈H¹(Σ) as a test function in (27) with the boundary condition (28). We get

0 = Z

Σ

Kf,τ

µ |∇τΠfp|²bfds+h[[v^p·ν]],Πfpi

H⁻¹²(Σ),H¹²(Σ). Adding the previous two equalities and using the definition (16) of Πfp, it follows Z

Ω

Km

µ |∇p|²dx+ Z

Σ

Kf,τ

µ |∇τΠfp|²bfds+ Z

Σ

Kf,ν

µ

[[p]]

bf

2

bfds+(2ξ−1)µ 4Kf,ν

Z

Σ

[[v^p·ν]]²bfds= 0.

We conclude, since ξ≥ ¹₂, and using (25), that the unique solutionp∈ W of (24)-(30) isp= 0, which proves that problem (17)-(22) has at most one solution inW.

Remark 3.6. Existence and uniqueness of a solution to problem (17)-(22) is proved for a non-immersed fracture (that is when ˜Ω\Σ is not connected) and in the caseξ > ¹₂ in [22] by using a mixed formulation. They obtained the result by showing an ellipticity property as well as an inf-sup inequality for this problem. This ellipticity property is no longer satisfied in the case where ξ= ¹₂ and their numerical results, based on the mixed finite element method, show instabilities in the limit ξ→ ¹₂.

Nevertheless, the existence result for this problem forξ= ¹₂ can be proved by using the standard variational formulation satisfied by the pressure in the porous matrix. In the case wherep^D= 0 (if not one hase to consider a lift of the Dirichlet data to the whole domain), the formulation reads:

Findp∈ W0 such thata(p, q) =L(q) for allq∈ W0, whereW0=W ∩γ0,D−1({0}), andais the bilinear form onW0defined by

a(p, q) = Z

Ω

Km

µ ∇p· ∇q dx+ Z

Σ

Kf,τ

µ ∇τΠfp∇τΠfq bfds+ Z

Σ

Kf,ν

µ [[p]]

bf

[[q]]

bf

bfds

andLis the linear form

L(q) = Z

Ω

Qq dx+ Z

Σ

bfQfΠfq ds.

Sinceξ= 1/2,W0 is a Hilbert space (see Remark 3.4). The bilinear formais then continuous and coercive onW0and the linear formLis continuous onW0so that the Lax-Milgram theorem applies. We can check that the solutionpto this variational formulation actually solves the problem under study (see the end of the proof of Theorem 4.11)

4. Finite volume scheme for the asymptotic model (P

)

We recall that we consider in this paper the case whereKmis isotropic. Under this assumption the framework of cell-centered finite volume method on the so-called orthogonal meshes is well adapted.

4.1. Notation and assumptions for the polygonal mesh

Let us define the notation we will use to describe and analyze our finite volume scheme. Most of the notation is inspired by that in [16], which is our reference for a general description and analysis of finite volume schemes for standard elliptic equations.

A mesh of the fractured domain Ω∪Σ is denoted by T = (M,S) whereM(resp. S) is a family of disjoint 2-dimensional control volumesK⊂Ω (resp. 1-dimensional control volumesσ⊂Σ).

• The control volumes ^K ∈ M are open convex polygons such that Ω =∪K∈MK. For any (^K,^L) ∈ M² with^K6=^L, either^K∩^L=∅, a vertex, or^K∩^L=σfor some edgeσ≡^K|^L.

Let Eint denote the set of interior edgesσ =^K|^L ⊂Ω andE_ext^D , E_ext^N the sets of edges lying on the boundary Γ with σ⊂ΓD or σ⊂ΓN respectively. The setE of all the edges can then be decomposed intoE =Eint∪ E_ext^D ∪ E_ext^N ∪S.

For each^K∈M, a discretization pointxK∈Kis chosen such that the segment [xK, xL] is orthogonal at the pointxσ to each edgeσ=K|L. This condition is very classical in the framework of cell-centered finite volume schemes for elliptic problems (see [16]). Such meshes are called orthogonal admissible meshes. For a mesh composed by triangles and satisfying the Delaunay condition, it is enough to choosexK to be the circumcenter ofK.

LetdK,σ>0 be the distance from xK toσ, and dK,L=dK,σ+dL,σ the distance betweenxKandxL. The set of edges of ^Kis denoted byEK,nK is the outward unit normal of^K, and for each edgeσ∈ EK

we will denote more preciselynK,σthe value ofnKalongσ. Finally for neighbor control volumes^Kand

L,nK,Lis the unit normal of ^Koriented from^Kto^L.

• We assume that the meshesMandSare compatible, that is for any control volumeσ∈Sthere exists (^K+_σ,^K−_σ)∈M²such thatσ=^K+_σ|^K−_σ with^K+_σ ⊂Ω⁺and ^K−_σ ⊂Ω⁻.

We denote byV the set of the verticeseof the meshS and by V_int the set of such vertices which are not on the boundary∂Σ (see Figure 2), so that we haveV=V_int∪ {∂Σ⁺, ∂Σ⁻}. For eachσ∈S, letVσ be the set of vertices inVbelonging to∂σ.

To each point e = σ|σ^′ ∈ V_int we associate the segment Se = [xσ, xσ^′] and the unit vector τ_σ,σ′

pointing from xσ towardsxσ^′. For e ∈ {∂Σ⁻, ∂Σ⁺} we note Se = [xσ, e] where σ ∈S is the unique element ofS such thate∈∂σ. We noteI= (Se)e∈V the set of such segments.

σ∈S p_σ+

pσ

p_σ−

dK−σ ,σ

pK−σ

Σ Se

σ^′ σ

dK+ σ ,σ

pK+ σ K+

σ

K− σ

xK−σ

e=σ|σ^′∈V n_K+

σ,σ

xσ x_σ′

xK+ σ

Figure 2. Geometry of the meshes along Σ

For each ^K ∈M or σ∈ S, m(^K) and m(σ) denote the 2D-measure of ^K, resp. the 1D-measure ofσ. The mesh size is defined by : size(T) = sup{diam(^K), ^K∈M}.

Finally, let rK >0 be the largest numberr such that the ball of radiusr centered atxK is contained in ^K. The regularity of the mesh is then measured by the quantity

reg(T) = max

K∈M

diam(^K) rK

,

that we will require to be bounded when the size of the mesh tends to 0 in our convergence results.

4.2. The discrete spaces 4.2.1. The discrete unknowns

• The matrix pressure field and its traces:

We associate to the meshT a set of discrete unknownsp^T composed as follows p^T = (p^M, γ0,Np^T, γ⁺p^T, γ⁻p^T)∈E(T)^def=R^M×R^EextN ×R^S×R^S.

The unknown vector p^M = (pK)K∈M ∈ R^M contains the cell-centered unknowns on the meshM, the vectorγ0,Np^T = (pσ)_σ∈E^N

ext represents the boundary values of the pressure on the part of the boundary where Neumann boundary conditions will be imposed. Since we are going to consider possible jumps of the pressure across the fracture Σ, we need to consider two different discrete traces of the pressure on Σ denoted byγ⁺p^{T def}= (pσ⁺)σ∈S∈R^S andγ⁻p^{T def}= (pσ⁻)σ∈S ∈R^S. The jumps and the mean-value across Σ ofp^T is defined by [[p^T]] =γ⁺p^T −γ⁻p^T andp^T = (γ⁺p^T +γ⁻p^T)/2. We can finally define, the boundary value on Γ of any given p^M ∈R^T byγ0p^M = (pKσ)σ∈Eext ∈R^Eext, and its restriction on ΓDbyγ0,Dp^M= (pKσ)_σ∈Eext^D ∈R^EextD , where^Kσ is the unique control volume inMsuch that σ⊂∂^Kσ.

As usual, in order to state our convergence results, discrete functions are identified as piecewise constant functions as follows

p^M= X

K∈M

1lKpK, γ⁺p^T = X

σ∈S

1lσp_σ⁺, γ⁻p^T = X

σ∈S

1lσp_σ−,

γ0,Dp^M = X

σ∈E_ext^D

1lσpKσ, γ0,Np^T = X

σ∈E_ext^N

1lσpσ.

• The fracture pressure:

We associate to the meshSon Σ, a fracture pressure unknown p^S= ((pσ)σ∈S, p_∂Σ⁻, p_∂Σ⁺)∈E(S)^def=R^S×R×R,

where pσ is a value at the center xσ of the edge σ and p_∂Σ⁻, p_∂Σ⁺ the boundary values at the two extremities∂Σ⁻ and∂Σ⁺ of Σ. We associate top^Sa piecewise constant function on Σ still denoted by p^S and definedp^{S def}= P

σ∈S1lσpσ. Notice that the boundary values p_∂Σ⁻ and p_∂Σ⁺ do not enter this definition. Fig. 2 sums up the different unknowns introduced near the fracture.

4.2.2. Discrete gradient

Let us define the diamond cellsDσ forσ∈ Eint∪ Eext andDσ⁺,Dσ⁻ forσ∈ S as shown in Figure 3. For σ = ^K|^L ∈ Eint, Dσ is the quadrangle whose diagonals are σ and [xK, xL]. The set of such diamond cells is called D_int. For σ =Eext∩ EK, Dσ is the triangle defined by the point xK and the edge σ. The set of such diamond cells is called D_ext. Finally, for σ ∈ S, Dσ⁺ and Dσ⁻ are the two triangles defined by the edge σ and by the points x_K⁺_σ and x_K⁻_σ respectively. We note D_Σ+ = {D_σ⁺, σ ∈ S}, D_Σ− = {D_σ−, σ ∈ S} and D=D_int∪D_ext∪D_Σ+∪D_Σ−.

Dσ

xK

Dσ

x_K⁻_σ x_K⁺_σ

K

xK

K K+

σ

K− σ L

xL

xσ σ∈S

D_σ⁺ Dσ⁻

σ=^K|^L∈ Eint

σ∈ Eext

Figure 3. The diamond cells

Definition 4.1 (Discrete gradient onM). For anyp^T ∈E(T), we define the vector-valued function∇^M,pDp^T by

∇^M,pDp^{T def}= 2 X

Dσ∈Dint σ=K|L

1lDσ

pL−pK

dK,L

nK,L+ X

Dσ∈Dext σ⊂ΓD

1lDσ

p^D_σ −pKσ

dK,σ nKσ+ X

Dσ∈Dext σ⊂ΓN

1lDσ

pσ−pKσ

dK,σ nK,σ

+ X

D_σ+∈D

Σ+

1lD_σ+

pσ⁺−p_K⁺_σ d_K⁺

σ,σ

n_K⁺_σ + X

D_σ−∈D_Σ−

1lD_σ−

pσ⁻−p_K⁻_σ d_K−

σ,σ

n_K⁻_σ

,

(30)

wherep^D_σ = _m(σ)¹ R

σp^D(s)ds.

The definition of such a discrete gradient was first proposed in [15] in order to study some links between homogenisation and numerical schemes. Notice that the coefficient 2 in front of the formula (30) is in fact the dimension d = 2 of the problem we are studying. Its presence is due to the fact that only the part of the gradient along the normalnσis approximated on each diamond cell. As we will see in the proof of Lemma 4.9, this coefficient 2 actually appears to be necessary to reach the weak convergence of the discrete gradient toward the continuous one.

Definition 4.2 (Fluxes accross Σ). For any p^T ∈E(T), we introduce the two mass fluxes across each edgeσ in Sby

v^p_σ^T+·n^+def=−Km

µ

pσ⁺−p_K⁺_σ d_K⁺

σ,σ

!

, v^p_σ^T−·n^−def=−Km

µ

pσ⁻−p_K⁻_σ d_K−

σ,σ

! .

Finally, the jump and the mean-value of these fluxes are defined by







[[v^p_σ^T ·ν]]^def=−(v^p_σ^T+·n⁺+v^p_σ^T−·n⁻), v^pσ^T ·ν^def=−v^p_σ^T+·n⁺−v^p_σ^T−·n⁻

2 .

(31)

Definition 4.3(Discrete fracture pressure). For any matrix pressure fieldp^T ∈E(T), following (16), we define the discrete fracture pressure Π^Sp^T ∈E(S) associated to p^T by

Π^Sp^T = 1

2 γ⁺p^T +γ⁻p^T

+(2ξ−1)µ

4Kf,ν bf,σ[[v^pT ·ν]], wherebf,σ is the mean value ofbf onσ.

Definition 4.4(Discrete gradient onS). For any fracture pressure fieldp^S∈E(S), we define the real-valued function∇^Sp^S∈L²(Σ) by

∇^Sp^S= X

e∈Vint e=σ|σ′

1lSe

pσ^′−pσ

m(Se) (τ_σ,σ′·τ) + 1lS∂Σ⁺

p_∂Σ⁺−pσ⁺

m(S_∂Σ⁺) −1lS∂Σ⁻

p_∂Σ⁻−pσ⁻

m(S_∂Σ−) . 4.2.3. The discrete norms

We define onE(T) the discreteH¹(Ω) norm kp^Tk1,T =

kp^Mk²_0,Ω+k∇^M,pDp^Tk²_0,Ω¹₂ ,

and onE(S), the discreteH¹(Σ) norm

kp^Sk1,S= kp^Sk²_0,Σ+k∇^Sp^Sk²_0,Σ¹2

. Finally, we define onE(T) the discreteW norm by

kp^Tk_WT,pD =

kp^Tk²_1,T +kΠ^Sp^Tk²_1,S+ (2ξ−1)k[[v^pT ·ν]]k²_0,Σ¹₂

. (32)

Lemma 4.5(Discrete trace inequality). There exists C >0 depending onreg(T)such that for allp^T ∈E(T), we have

kγ0p^Mk0,Γ+kγ⁺p^Tk0,Σ+kγ⁻p^Tk0,Σ≤Ckp^Tk1,T.

Lemma 4.6 (Discrete Poincar´e Lemma). There existsC=C(Ω)such that for all p^T ∈E(T), we have kp^Mk0,Ω≤C

k∇^M,pDp^Tk0,Ω+kp^Dk¹

2,ΓD

.

The proofs of Lemmas 4.5 and 4.6 are direct adaptations of those for the analogous classical results given in [16].

4.3. The cell-centered numerical scheme for problem (PA) 4.3.1. Description of the scheme

• Flow in the porous matrix Ω. Integrating equation (17) over each control volumes ^K ∈ M, the classical cell-centered FV method reads

X

σ∈EK

FK,σ=m(^K)QK, ∀K∈M, (33)

whereQKis the mean value ofQon^K, andFK,σis the numerical flux approximatingR

σv^p·nKds. This numerical flux is defined by

FK,σ

def=−m(σ)Km

µ

pK,σ−pK

dK,σ

,

wherepK,σis an approximate value of the pressure on the side ofσtouching the control volume^K. Let us see how to determine pK,σ.